Group

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Group
1 r* i! a5 T* j! O$ ?A group is defined as a finite or infinite set of Operands
- s( j( B2 T! Q4 T4 t; [+ J/ E* n (called ``elements'') , , , ... that may be combined or ``multiplied'' via a Binary Operator
; l5 u! t- e: |" H3 o to form well-defined products and which furthermore satisfy the following conditions:
1. Closure: If and are two elements in , then the product is also in .
* e/ p) ]6 J* e2. Associativity: The defined multiplication is associative, i.e., for all , .
2 ?/ w) D2 o6 o& j, g& y/ o3. Identity: There is an Identity Element
% \3 s% c7 h8 y+ x (a.k.a. , , or ) such that for every element .
& ^- Q" @9 m" h# ?4. Inverse: There must be an inverse or reciprocal of each element. Therefore, the set must contain an element such that for each element of .
A group is therefore a Monoid
4 P* ~1 L" h" d8 Y, f+ t for which every element is invertible. A group must contain at least one element.
' n4 X  ^& y6 o. V9 E+ h% @
5 f6 _" M# x7 i, E6 JThe study of groups is known as Group Theory
. If there are a finite number of elements, the group is called a Finite Group
and the number of elements is called the Order
: Y" N: P( a. }# Y. M of the group.

$ D/ E+ b+ ?  q0 [1 K1 rSince each element , , , ..., , and is a member of the group, group property 1 requires that the product
; y; J# o5 V+ t

(1)

+ X0 f4 M* M0 b& D

must also be a member. Now apply to ,

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; A: t7 m+ ^# l' u9 B

(2) 7 K& k& _) y( P2 ~5 L/ H# C) ~2 X" B

6 G; @4 W6 g* z" h8 \But

1 Y* d! D( i. p# g7 u# p
6 G/ v1 P0 @7 ?, d, Q4 p, ~	(3)

so
6 n. I+ Y( l2 v  o+ K# p3 `

* c# q" u& Q$ _  k- j5 S& T

(4)

, n! Z+ q% w% y1 o
' K- h! u" j( J4 n
which means that

; F( G4 E( K8 E# n* B

(5) 7 Z- j' @7 H0 N4 k9 _6 N: e

; _/ X. p0 K, c# n& u3 o

  v. t# g- u0 z; @2 Mand
6 Y: J- \$ F; U: G& M

(6) " L2 ?7 D. u& g

3 O& u$ @) U2 |, |2 d
3 t& L9 v/ n# C+ E3 F
$ u. I* l6 C, C' A
0 Z4 x4 M4 m3 @0 X( y+ p) G

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